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What is the probability that a large random matrix has no real eigenvalues? (English) Zbl 1375.60019
Summary: We study the large-$$n$$ limit of the probability $$p_{2n,2k}$$ that a random $$2n \times 2n$$ matrix sampled from the real Ginibre ensemble has $$2k$$ real eigenvalues. We prove that $\lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n} \log p_{2n,2k} = \lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n} \log p_{2n,0} = - \frac{1}{\sqrt 2\pi} \zeta \left (\frac 32\right ) ,$ where $$\zeta$$ is the Riemann zeta-function. Moreover, for any sequence of nonnegative integers $$(k_n)_{n\geq 1}$$, $\lim_{n\longrightarrow \infty} \frac{1}{\sqrt 2n}\log p_{2n,2k_n} = - \frac{1}{\sqrt 2\pi} \zeta \left (\frac 32\right),$ provided $$\displaystyle \lim_{n\to\infty}\left (n^{-1/2}\log(n)\right)k_n=0$$.

##### MSC:
 60B20 Random matrices (probabilistic aspects) 60F10 Large deviations
##### Keywords:
real Ginibre ensemble; large deviations
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